Two-Scale G-Convergence of Integral Functionals and its Application to Homogenisation of Nonlinear High-Contrast Periodic Composites
Two-Scale G-Convergence of Integral Functionals and its Application to Homogenisation of Nonlinear High-Contrast Periodic Composites
复制标题
积分泛函的两尺度 G 收敛及其在非线性高对比度周期性复合材料均匀化中的应用
DOI：
10.1007/s00205-011-0481-4
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发表时间：
2011
影响因子：
2.5
通讯作者：
Cherdantsev M
中科院分区：
文献类型：
--
作者：
Cherdantsev M
一个分析框架的开发，通过均匀化极限（不一定是凸）的复合材料的材料特性振荡的小周期ε和表现出高对比度的阶本构，“应力-应变”，响应不同部分的周期细胞的变分问题。本文的方法基于“双尺度Γ-收敛”的概念，其是经典Γ-收敛（De Giorgi和Franzoni in Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur（8）58：842-850，1975）和更近的双尺度收敛（Nguetseng in SIAM J Math Anal 20：608-623，1989）的一种“混合”。本研究的重点是一个基本的高对比度模型，其中“软”夹杂物嵌入在一个“硬”矩阵。证明了在Lp-空间中标准的Γ-收敛不能给出正确的极限问题，这是由于极小化序列缺乏Lp-紧性.使用适当的双尺度紧性陈述作为替代的出发点，通过经典齐次化、拟凸函数理论和多尺度分析的技术的组合，确定了原泛函族的双尺度Γ-极限。相关的结果可以被认为是Müller（Arch Rational Mech Anal 99：189-212，1987）的著名定理的“非经典”双尺度扩展。
An analytical framework is developed for passing to the homogenisation limit in (not necessarily convex) variational problems for composites whose material properties oscillate with a small periodεand that exhibit high contrast of orderbetween the constitutive, “stress-strain”, response on different parts of the period cell. The approach of this article is based on the concept of “two-scaleΓ-convergence”, which is a kind of “hybrid” of the classicalΓ-convergence (De Giorgi and Franzoni in Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur (8)58:842–850, 1975) and the more recent two-scale convergence (Nguetseng in SIAM J Math Anal 20:608–623, 1989). The present study focuses on a basic high-contrast model, where “soft” inclusions are embedded in a “stiff” matrix. It is shown that the standardΓ-convergence in theLp-space fails to yield the correct limit problem asdue to the underlying lack ofLp-compactness for minimising sequences. Using an appropriate two-scale compactness statement as an alternative starting point, the two-scaleΓ-limit of the original family of functionals is determined via a combination of techniques from classical homogenisation, the theory of quasiconvex functions and multiscale analysis. The related result can be thought of as a “non-classical” two-scale extension of the well-known theorem by Müller (Arch Rational Mech Anal 99:189–212, 1987).